Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 226, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ENERGY DECAY OF A VARIABLE-COEFFICIENT WAVE
EQUATION WITH NONLINEAR TIME-DEPENDENT
LOCALIZED DAMPING
JIEQIONG WU, FEI FENG, SHUGEN CHAI
Abstract. We study the energy decay for the Cauchy problem of the wave
equation with nonlinear time-dependent and space-dependent damping. The
damping is localized in a bounded domain and near infinity, and the principal
part of the wave equation has a variable-coefficient. We apply the multiplier
method for variable-coefficient equations, and obtain an energy decay that
depends on the property of the coefficient of the damping term.
1. Introduction
Let n ≥ 2 be an integer. We consider the energy decay for the solution to the
Cauchy problem of the wave equation with time-dependent and space-dependent
damping,
utt − div(A(x)∇u) + u + σ(t)ρ(x, ut ) = 0,
u(0, x) = u0 (x),
ut (0, x) = u1 (x),
x ∈ Rn , t > 0,
n
x∈R ,
(1.1)
(1.2)
n
where A(x) = (aij (x)) is a symmetric, positive matrix for each x ∈ R ; σ : R+ →
R+ is a non-increasing function of class C 1 . Let Ω and Ω1 be two bounded domains
such that Ω1 ⊂ Ω. We assume that
(
a(x)ut ,
x ∈ R n \ Ω1
ρ(x, ut ) =
(1.3)
a(x)h(ut ), x ∈ Ω1
where h : R → R is a nonlinear continuous nondecreasing function satisfying
h(s)s > 0 for all s 6= 0 and a(·) ∈ L∞ (Rn ) is a nonnegative function satisfying
x ∈ Ω1 ∪ Ωc .
a(x) ≥ ε0 > 0,
(1.4)
When x ∈ Ω1 ,
c1 |ut |m ≤ |h(ut )| ≤ c2 |ut |1/m ,
c3 |ut | ≤ |h(ut )| ≤ c4 |ut |,
|ut | ≤ 1,
|ut | ≥ 1
where m ≥ 1 and ci > 0 (i = 1, 2, 3, 4) are given numbers.
2010 Mathematics Subject Classification. 35L05, 35L70, 93B27.
Key words and phrases. Energy decay; time-dependent and space-dependent damping;
localized damping; Riemannian geometry method; variable coefficient.
c
2015
Texas State University - San Marcos.
Submitted May 28, 2015. Published September 2, 2015.
1
(1.5)
(1.6)
2
J. WU, F. FENG, S. CHAI
EJDE-2015/226
Since (1.4) implies that the dissipation may vanish in Ω\Ω1 , we call the damping
σ(t)ρ(x, ut ) a time-dependent localized damping.
Many studies concerning energy decay of wave equations in the bounded domain
are available in the literature. We refer the readers to [3, 8, 9, 16]. The author of
[8] derived precise energy decay estimates for the initial-boundary value problem
to the wave equation with a localized nonlinear dissipation which depended on the
time as well as the space variable. The result of [8] is a generalization of the work
[9], where the dissipation was independent of the time. A wave equation with timedependent but space-independent damping was investigated in [3] and the decay
rate was obtained by solving a nonlinear ODE.
For the energy decay of wave equation in the exterior domain or on the whole
space, many results have been obtained for the case of constant coefficients, that
is A(x) ≡ I (see [1, 2, 4, 10, 11, 12, 17]). The damping in [4, 11, 17] were timeindependent and localized in a sub-domain of Rn . Energy decay of the Cauchy
problem for the wave equation with time-dependent damping was studied in [2]
where the coefficient of the damping did not depend on the space variable. It is
interesting to study the case where the damping is time-dependent and exists on
a local domain of Rn . [1] addressed the decay for a wave equation on an exterior
domain where the damping is time-dependent and effective only in a ball of Rn .
The dimension of the space variable was restricted to be odd since the Lax-Phillips’
scattering theory was used in [1]. Nakao [10] considered the Cauchy problem of wave
equation with a dissipation effective outside of a given ball of Rn .
However, in the case of variable coefficient, few results about the energy decay of
wave equation in the exterior domain or on the whole space can be seen (see[5, 13]).
Yao [13] studied the energy decay for the Cauchy problem of the variable-coefficient
wave equation with a linear damping. The authors of [5] considered the energy
decay of variable-coefficient wave equation with nonlinear damping in an exterior
domain.
The purpose of this paper is to derive the energy decay for (1.1)-(1.2) with the
assumptions (1.3)-(1.6). Our first main goal is to dispense with the restriction
A(x) ≡ I. We will use the Riemannian geometry method which was introduced
by Yao [14] (see also [15]) to deal with controllability for the wave equation with
variable-coefficient principal part and has been viewed as an important method for
variable-coefficient models.
Our second goal is to generalize the work [2], where the damping exists on the
whole space. In this paper, we assume that the damping is localized near the origin
point and near infinity and may disappear in a large area. We try to explain how the
time-dependence of the damping affect the decay when the dissipation is effective
in a localized domain. Energy decay is obtained under some conditions on A(x)
and the coefficient of the damping.
We introduce a Riemanian metric on Rn by
g(x) = A−1 (x)
for
x ∈ Rn .
(1.7)
We denote by ∇f and ∇g f the gradients of f in the standard metric of the Euclidean
space Rn and in the metric g, respectively.
The energy of the model (1.1)-(1.2) is defined by
Z
1
(u2 + |∇g u|2g + u2 )dx.
(1.8)
E(t) =
2 Rn t
EJDE-2015/226
ENERGY DECAY
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where ∇g u = A(x)∇u and
|∇g u|2g = h∇g u, ∇g uig = hA(x)∇u, ∇ui =
n
X
aij (x)
ij=1
∂u ∂u
.
∂xj ∂xi
We refer the readers to [14] and [15] for more information about the metric g(x).
We use the following assumption in this article
(H1) There is a vector field H on (Rn , g) such that
Dg H(X, X) ≥ σ̄|X|2g ,
X ∈ Rnx , x ∈ Ω̄,
(1.9)
where σ̄ > 0 is a constant and Dg is the Levi-Civita connection of the
metric g.
Our main result read as follows.
Theorem 1.1. Let (u0 , u1 ) ∈ H 1 (Rn ) × L2 (Rn ). In addition to (H1) assume that
σ : R+ → R+ is a non-increasing C 1 function and σ(t) ≥ σ0 > 0 for all t ≥ 0.
Then (1.1)-(1.2) admits a unique solution u satisfying
2
m−1
1
E(t) ≤ CE(0) R t
, ∀t > 0 if m > 1,
σ(s)ds
0
and
Z
E(t) ≤ CE(0) exp(−ω
t
σ(s)ds),
∀t > 0
if m = 1,
0
for some ω > 0 and C > 0.
2. Proof of main result
To prove our main result we use the following lemma.
Lemma 2.1 ([6]). Let E : R+ → R+ be a non-increasing function and φ : R+ → R+
a strictly increasing C 1 function such that
φ(0) = 0,
φ(t) → +∞
as t → +∞.
Assume that there exist r ≥ 0 and ω > 0 such that
Z +∞
1
E 1+r (t)φ0 (t)dt ≤ E r (0)E(S),
ω
S
0 ≤ S < +∞.
Then
E(t) ≤ E(0)
1 + r 1/r
1 + ωrφ(t)
E(t) ≤ CE(0)e1−ωφ(t)
for all t ≥ 0, if r > 0,
for all t ≥ 0, if r = 0.
To apply Lemma 2.1, we need some estimates on the energy terms, which are
based on the identities below. Multiply (1.1) by ut and integrate by parts over Rn
with respect to the variable x to obtain
Z
d
E(t) = −
σ(t)ρ(x, ut )ut dx.
(2.1)
dt
Rn
4
J. WU, F. FENG, S. CHAI
EJDE-2015/226
Let H be a vector field on Rn . Multiply (1.1) by H(u) and integrate by parts
over Rn with respect to the variable x to obtain
Z
Z
d
ut H(u)dx +
Dg H(∇g u, ∇g u)dx
dt Rn
Rn
Z
Z
1
+
H(u)u dx +
(u2 − |∇g u|2g )divHdx
(2.2)
2 Rn t
n
R
Z
=−
σ(t)ρ(x, ut )H(u)dx.
Rn
Let q ∈ C 1 (Rn ) be a function. Multiply (1.1) by qu and integrate by parts over
R with respect to the variable x to obtain
Z
Z
Z
Z
d
2
2
2
u div A∇q dx +
quut dx +
q |∇g u|g − ut dx −
qu2 dx
dt Rn
Rn
Rn
Rn
Z
(2.3)
=−
quσ(t)ρ(x, ut )dx.
n
Rn
Proof of Theorem 1.1. Let the vector field H be such that the assumption (H1)
ˆ
¯
ˆ
holds. Let Ω̂ ⊂ Rn and Ω̂ ⊂ Rn be two bounded domains such that Ω̄ ⊂ Ω̂ ⊂ Ω̂ ⊂ Ω̂.
Let ϕ and ψ be two C0∞ functions and 0 ≤ ϕ ≤ 1, 0 ≤ ψ ≤ 1, and
(
(
1, x ∈ Ω̂,
1, x ∈ Ω,
ϕ=
ψ=
(2.4)
ˆ
c
n
0, x ∈ Ω̂ = R \ Ω̂,
0, x ∈ Rn \ Ω̂.
Let q0 = div(ϕH)
− σ̄ϕ. Replacing H by ϕH in (2.2) and replacing q by q0 in (2.3),
2
respectively, we can obtain two identities. Then we add them up and obtain
Z
Z
d
ut ϕH(u) + q0 uut dx +
Dg (ϕH)(∇g u, ∇g u)dx
dt Rn
Rn
Z
Z
+
ϕH(u)u dx +
σ̄ϕ(u2t − |∇g u|2g )dx
n
n
R
ZR
Z
(2.5)
2
−
|u| div(∇g q0 )dx +
q0 u2 dx
Rn
Rn
Z
Z
=−
q0 uσ(t)ρ(x, ut )dx −
σ(t)ρ(x, ut )ϕH(u)dx.
Rn
Rn
Let k be a large constant determined later. Combining (2.1) with (2.5), we have
Z
Z
d
ut ϕH(u) + q0 uut dx + kE 0 (t) +
Dg (ϕH)(∇g u, ∇g u)dx
dt Rn
Rn
Z
Z
Z
2
2
+
ϕH(u)u dx +
σ̄ϕ(ut − |∇g u|g )dx −
|u|2 div(∇g q0 )dx
Rn
Rn
Rn
Z
+
q0 u2 dx
(2.6)
n
R
Z
Z
=−
q0 uσ(t)ρ(x, ut )dx −
σ(t)ρ(x, ut )ϕH(u)dx
Rn
Rn
Z
−k
σ(t)ρ(x, ut )ut dx.
Rn
EJDE-2015/226
ENERGY DECAY
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Set
Z
Z
Dg (ϕH)(∇g u, ∇g u)dx+
Y (t) =
Rn
Rn
σ̄ϕ(u2t −|∇g u|2g )dx+k
Z
σ(t)ρ(x, ut )ut dx.
Rn
Set σ1 = supΩ̂¯ |Dg (ϕH)(X, X)|. Noticing (1.3),(1.4),(1.9) and (2.4), we have the
following estimates on Y (t):
Z
Z
Z
Dg (ϕ)H(∇g u, ∇g u)dx +
σ̄ϕu2t dx −
σ̄ϕ|∇g u|2g dx
n
Ω̂\Ω
R \Ω
Ω̂\Ω
Z
Z
Z
3k
k
2
2
0 σ(t)ut dx +
σ̄ut dx +
σ(t)ρ(x, ut )ut dx
+
4 Rn \Ω
4 Rn
Ω
Z
Z
k
2
ε0 σ(t) + σ̄ϕ u2t dx
≥ (−σ1 − σ̄)
|∇g u|g dx +
4
n
Ω̂\Ω
R \Ω
Z
Z
3k
+
σ̄u2t dx +
σ(t)ρ(x, ut )ut dx .
4 Rn
Ω
Y (t) ≥
Noticing σ(t) ≥ σ0 > 0, we choose k large enough so that
Z
Y (t) ≥
Z
2
σ̄|ut | dx −
Rn
(σ1 +
σ̄)|∇g u|2g
Ω̂\Ω
3k
+
4
kσ0 ε0
4
(2.7)
> σ̄. Then we have
Z
σ(t)ρ(x, ut )ut dx .
(2.8)
Rn
Now we estimate the last term of the right side of (2.8).
Z
σ(t)ρ(x, ut )ut dx
Rn
Z
Z
≥
ψσ(t)ρ(x, ut )ut dx
ψσ(t)ρ(x, ut )ut dx =
ˆ
Z Ω̂
=
ψσ(t)a(x)u2t dx +
ψσ(t)ρ(x, ut )ut dx
ˆ
Ω̂\Ω1
Ω1
Z
Z
Z
=
ψσ(t)a(x)u2t dx −
σ(t)a(x)u2t dx +
ψσ(t)ρ(x, ut )ut dx .
n
ZR
ˆ
Ω̂
Ω1
(2.9)
Ω1
For the first term on the right of (2.9), noticing σ(t) ≥ σ0 and using (1.4), (2.4)
and (2.3) for q = σ0 ψa(x), we have
Z
ˆ
Ω̂
ψσ(t)a(x)u2t dx ≥ σ0
Z
Z
σ0 ψa(x)u2t dx
Z
Z
d
≥
σ0 a(x)ψuut dx + σ0 ε0
|∇g u|2g dx
dt Rn
Ω̂\Ω
Z
Z
2
− σ0
u div A∇ a(x)ψ dx + σ0
a(x)ψu2 dx
Rn
Rn
Z
+ σ0
a(x)ψσ(t)ρ(x, ut )u dx.
ˆ
Ω̂
ψa(x)u2t dx =
Rn
Rn
ˆ
since supp ψ ⊂ Ω̂ and a(x) ≥ ε0 > 0 for x ∈ Ωc .
(2.10)
6
J. WU, F. FENG, S. CHAI
Combining (2.8) with (2.9) and (2.10), we have
Z
kσ ε
Z
0 0
2
Y (t) ≥
− (σ1 + σ̄)
|∇g u|g dx +
σ̄u2t dx
4
Ω̂\Ω
Rn
Z
Z
σ0 k h d
k
σ(t)ρ(x, ut )ut dx +
a(x)ψuut dx
+
2 n
4 dt Rn
Z R
Z
−
u2 div A∇ a(x)ψ dx +
a(x)ψu2 dx
n
n
R
ZR
i kZ
σ(t)a(x)u2t dx
+
a(x)ψσ(t)ρ(x, ut )udx −
4 Ω1
Rn
Z
k
+
σ(t)ρ(x, ut )ut dx
4 Ω1
EJDE-2015/226
(2.11)
Choose k large enough so that kσ40 ε0 > σ̄ + σ1 . Then it follows from (2.11) and
(2.6) that
Z
Z
σ0 k
d
ut ϕH(u) + q0 uut dx + kE(t) +
a(x)ψuut dx
dt Rn
4 Rn
Z
Z
Z
k
k
2
+
σ̄ut dx +
σ(t)ρ(x, ut )ut dx −
σ(t)a(x)ψu2t dx
2 Ω1
4 Ω1
Rn
Z
Z
k
σ0 k h
+
σ(t)ρ(x, ut )ut dx +
aψ − div A∇ a(x)ψ) u2 dx
4 Ω1
4
n
R
(2.12)
Z
i Z
+
a(x)ψσ(t)ρ(x, ut )udx +
ϕH(u)u dx
n
n
ZR
Z R
+
q0 − div ∇g q0 u2 dx +
q0 uσ(t)ρ(x, ut )dx
n
Rn
ZR
+
σ(t)ρ(x, ut )ϕH(u)dx ≤ 0.
Rn
We may have equality by setting q = σ̄2 in (2.3). Then add that equality to (2.12)
to yield
Z
Z
σ0 k
d
ut ϕH(u) + q0 uut dx + kE(t) +
a(x)ψuut dx
dt Rn
4 Rn
Z
Z
σ̄
σ̄
uut dx +
|∇g u|2g + u2t + u2 + σ(t)ρ(x, ut )u dx
+
Rn 2
Rn 2
Z
Z
k
k
+
σ(t)ρ(x, ut )ut dx −
σ(t)a(x)ψu2t dx
2 Ω1
4 Ω1
Z
Z k
σ0 k h
+
σ(t)ρ(x, ut )ut dx +
aψ − div A∇ a(x)ψ u2 dx
(2.13)
4
4
Rn
Z Ω1
Z
i
+
a(x)ψσ(t)ρ(x, ut )udx +
ϕH(u)u dx
Rn
Rn
Z
Z
+
q0 − div ∇g q0 u2 dx +
q0 σ(t)ρ(x, ut )u dx
n
Rn
ZR
+
σ(t)ρ(x, ut )ϕH(u)dx ≤ 0.
Rn
EJDE-2015/226
ENERGY DECAY
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Set
Z
X(t) =
Rn
σ0 k
ut ϕH(u) + q0 uut dx + kE(t) +
4
Z
Z
a(x)ψuut dx +
Rn
Rn
σ̄
uut dx.
2
From (2.13) and the definition of E(t), we have
σ̄E(t)
Z
Z
σ̄
d
σ(t)ρ(x, ut )u dx −
σ(t)ρ(x, ut )ϕH(u) dx
≤ − X(t) −
dt
Rn 2
Rn
Z
Z
k
−
σ(t)ρ(x, ut )ut dx
q0 σ(t)ρ(x, ut )udx −
2 Rn
n
R
Z
Z σ0 k h
k
aψ − div A∇ a(x)ψ u2 dx
σ(t)ρ(x, ut )ut dx −
−
4 Ω1
4
Rn
Z
i kZ
+
a(x)ψσ(t)ρ(x, ut )u dx +
σ(t)a(x)ψu2t dx
4 Ω1
Rn
Z
Z
−
ϕH(u)u dx −
q0 − div ∇g q0 u2 dx
Rn
Rn
k
Z
d
σ(t)ρ(x, ut )u dx − E 0 (t)
≤ − X(t) + σ4 dt
2
n
Z
ZR
Z
σ0 kσ2
k
+(
σ(t)a(x)u2t dx + σ5
|∇g u|g u dx
+ σ3 )
|u|2 dx +
ˆ
4
4 Ω1
Ω̂
Ω̂
Z
+ σ5
σ(t)|ρ(x, ut )||∇g u|g dx
(2.14)
Rn
where
σ2 = sup |aψ − div ∇g (aψ)|,
¯
ˆ
x∈Ω̂
σ4 = max
σ3 = sup |q0 − div ∇g q0 |,
σ0 k|a|∞ σ̄
, , sup |q0 | ,
4
2 Ω̂¯
¯
x∈Ω̂
σ5 = sup |ϕH|.
¯
Ω̂
For the second term, the sixth term and the last term on the right of (2.14), we
use Young’s inequality to obtain
Z
Z
σ4 σ(t)ρ(x, ut )udx ≤ σ4
Cε |σ(t)ρ(x, ut )|2 + εu2 dx
n
Rn
ZR
(2.15)
≤ σ4
Cε |σ(t)ρ(x, ut )|2 + 2εC4 E(t),
Rn
Z
Z
Z
ε|∇g u|2g dx + Cε σ5
u2 dx,
Rn
Ω̂
Z
≤ 2εC5 E(t) + Cε σ5
u2 dx,
|∇g u|g udx ≤ σ5
σ5
Ω̂
(2.16)
Ω̂
and
Z
σ5
σ(t)|ρ(x, ut )||∇g u|g dx
Z
Z
≤ εσ5
|∇g u|2g dx + Cε σ5
|σ(t)ρ(x, ut )|2 dx
Rn
Rn
Z
≤ 2εC5 E(t) + Cε σ5
|σ(t)ρ(x, ut )|2 dx
Rn
Rn
(2.17)
8
J. WU, F. FENG, S. CHAI
EJDE-2015/226
where ε > 0 is small and Cε is a constant dependent on ε.
Let φ be a function satisfying the conditions in Lemma 2.1. Let r ≥ 0 be a
constant determined later. Multiplying (2.14) by E r φ0 and integrating on [S, T ]
with respect to t, from (2.15) -(2.17) we have
Z
T
E r+1 (t)φ0 σ̄dt
S
T
Z
r 0
≤−
Z
0
T
r 0
E φ X (t)dt + (σ4 Cε + σ5 Cε )
S
Z
T
E r+1 φ0 dt −
+ (2εσ4 + 4εσ5 )
S
k
2
Z
Z
Z
|σ(t)ρ(x, ut )|2 dx dt
E φ
Rn
S
T
E r φ0 E 0 dt
(2.18)
S
Z T
kσ0 σ2
u2 dx dt
+ σ3 + C ε σ5 )
E r φ0
ˆ
4
Ω̂
S
Z
Z
k T r 0
E φ
σ(t)a(x)u2t dx dt.
+
4 S
Ω1
+(
Using a similar argument as in [13] and [7], we can prove that if T − S is large
enough, then
Z
T
E r φ0
S
Z
ˆ
Ω̂
u2 dx dt ≤ Cη
T
Z
E r φ0
Z
|σ(t)ρ(x, ut )|2 dx
Rn
S
Z
Z
σ(t)u2t dx dt + η
+
Ω1
(2.19)
T
E r+1 (t)φ0 dt
S
holds for any η > 0.
Rt
For the rest of this article, let φ(t) = 0 σ(s)ds. It is clear that φ(t) satisfies the
conditions of Lemma 2.1. Noticing that |X(t)| ≤ CE(t) and E 0 (t) ≤ 0, we have by
integrating by parts with respect to t,
Z
T
−
E r φ0 X 0 (t)dt
S
T Z T
= −E (t)φ (t)X(t) +
(E r φ0 )0 X(t)dt
S
S
Z T
Z
r+1
r−1 0 0
≤ Cσ(0)E
(S) +
|rE
E φ X(t)|dt +
r
0
S
≤ Cσ(0)E r+1 (S) + Cσ(0)
Z
T
S
rE r (−E 0 )dt +
Z
T
|E r φ00 X(t)|dt
(2.20)
S
T
E r+1 (−φ00 )dt
S
≤ CE r+1 (S).
We also have
−
k
2
Z
T
E r φ0 E 0 dt ≤ CE r+1 (S).
(2.21)
S
Now we estimate the second term on the right of (2.18). Set Ω+
1 = {x ∈ Ω1 ; |ut | ≥
+
−
1} and Ω−
=
{x
∈
Ω
;
|u
|
<
1}.
Then
Ω
=
Ω
∪
Ω
.
Noticing
(1.5), using Hölder
1
t
1
1
1
1
EJDE-2015/226
ENERGY DECAY
9
inequality and Young’s inequality, we have
Z T
Z
E r φ0
|σ(t)ρ(x, ut )|2 dx dt
Ω−
1
S
T
Z
r 0
=
Z
σ 2 (t)|a(x)h(ut )|2 dx dt
E φ
Ω−
1
S
Z
T
E r φ0 σ 2 (t)
≤ c2
Z
Ω−
1
S
Z
a2 (x) h(ut )ut
T
Z
E r φ0
= c2
Ω−
1
S
T
Z
2
2− m+1
σ(t)a(x)h(ut )ut
2
m+1
Z
2
E r φ0 σ(t)2− m+1
≤ C|a|∞
Ω−
1
S
2
2− m+1
≤ C|a|∞
2
m+1
h
σ 2 (0) ε1
T
Z
dx dt
2
2− m+1
a(x)σ(t)
dx dt
(2.22)
2
m+1
dt
σ(t)a(x)h(ut )ut dx
2
E r φ0 σ(t)− m+1
m+1
m−1
dt
S
T
Z
Z
+ Cε1
S
Ω−
1
Now, we set r =
m−1
2 .
T
Z
i
σ(t)a(x)h(ut )ut dx dt .
Thus, we have from (2.22) and (2.1)
Z
r 0
|σ(t)ρ(x, ut )|2 dx dt
E φ
Ω−
1
S
2− 2
C|a|∞ m+1 σ 2 (0)
h
≤
Z
(2.23)
T
ε1
E
m+1
2
0
φ dt + Cε1 E(S) .
S
Noticing (1.6) and (2.1), it is easy to obtain
Z
Z T
Z
|σ(t)ρ(x, ut )|2 dx dt ≤ c4 σ(0)|a|∞
E r φ0
Ω+
1
S
T
E r φ0 (−E 0 (t))dt
S
≤ CE
r+1
(2.24)
(S).
On the other hand, it is easy to obtain
Z T
Z
E r φ0
|σ(t)ρ(x, ut )|2 dx dt ≤ CE r+1 (S)
(2.25)
Rn \Ω1
S
since ρ(x, ut ) = a(x)ut when x ∈ Rn \ Ω1 .
From (2.23)-(2.25), we have
Z T
Z
E r φ0
|σ(t)ρ(x, ut )|2 dx dt
Rn
S
≤
2− 2
Cε1 |a|∞ m+1 σ 2 (0)
Z
(2.26)
T
E
m+1
2
0
φ dt + Cε1 E(S) + CE
r+1
(S).
S
Similarly, for the last term on the right side of (2.18), we have
Z T
Z
E r φ0
σ(t)a(x)u2t dx dt
S
Ω1
m−1
m+1
Z
(2.27)
T
≤ Cε1 |a|∞ σ(0)
E
S
m+1
2
0
φ dt + Cε1 E(S) + CE
r+1
(S).
10
J. WU, F. FENG, S. CHAI
EJDE-2015/226
At this point, we choose ε and η small enough so that
kσ0 σ2
σ̄
2εσ4 + 4εσ5 + η(
+ σ3 + Cε σ5 ) < .
4
2
Then using (2.19)-(2.21), (2.18) becomes
Z
σ̄ T r+1
E
(t)φ0 dt
2 S
kσ0 σ2
≤ CE r+1 (S) + σ4 Cε + σ5 Cε + Cη (
+ σ3 + Cε σ5 )
4
Z
Z T
Z T
k
r 0
2
E φ
|σ(t)ρ(x, ut )| dx dt −
×
E r φ0 E 0 dt
2 S
S
Rn
Z
Z
T r 0
k
kσ0 σ2
+
E φ
σ(t)a(x)u2t dx dt.
+ Cη (
+ σ3 + Cε σ5 )
4
4
S
Ω1
(2.28)
Once ε and η are fixed, we pick ε1 small enough so that
m−1
σ̄
kσ0 σ2
k
m+1
+ σ3 + C ε σ5 ) ≤ .
Cε1 |a|∞
σ(0) + σ 2 (0) (σ4 Cε + σ5 Cε + + Cη (
4
4
4
Then using (2.26)-(2.27) in (2.28), we have
Z T
m+1
(2.29)
E 2 φ0 dt ≤ CE(S).
S
Let T → ∞ in (2.29). Then Theorem 1.1 follows from Lemma 2.1 and (2.29) with
Rt
and φ(t) = 0 σ(s)ds.
r = m−1
2
Acknowlegments. This work was supported by the National Science Foundation
of China for the Youth (61104129,11401351), by the Youth Science Foundation
of Shanxi Province (2011021002-1), by the National Science Foundation of China
(11171195, 61174082 and 61174083).
References
[1] A. Bchatnia, M. Daoulatli; Local energy decay for the wave equation with a nonlinear timedependent damping, Applicable Analysis, 92(11) (2013), 2288-2308.
[2] A. Benaissa, S. Benazzouz; Energy decay of solutions to the Cauchy problem for a nondissipative wave equation, Journal of Mathematical Physics 51, 123504 (2010)
[3] M. Daoulatli; Rates of decay for the wave systems with time dependent damping, Discrete
and Continuous Dynamical systems, 31(2) (2011), 407-443.
[4] M. Daoulatli, B. Dehman, M. Khenissi; Local energy decay for the elastic system with nonlinear damping in an exterior domain, SIAM J Control. Optim., 48(8) (2010), 5254-5275.
[5] Jing Li, Shugen Chai, Jieqiong Wu; Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain, Z. Angew. Math. Phys.
66 (2015) 95-112.
[6] P. Martinez; A new method to obtain decay rate estimates for dissipative systems, ESAIM
Control Optim. Calc. Var. 4, (1999), 419-444.
[7] M. Nakao, I. H. Jung; Energy decay for the wave equation in exterior domains with some
half-linear dissipation, Differential and Integral Equations, 18(8) (2003), 927-948.
[8] M. Nakao; On the decay of solutions of the wave equation with a local time-dependent nonlinear dissipation, Advance in Mathematical and Applications Gakkotosho, Tokyo, 7(1) (1997),
317-331.
[9] M. Nakao; Decay of solutions of the wave equation with a local nonlinear dissipation, Math.
Ann. 305 (3) (1996), 403-417.
[10] M. Nakao; Decay of solutions to the Cauchy problem for the Klein-Gordon equation with a
localized nonlinear dissipation, Hokkaido Mathematical Journal, 27 (1998) 245-271.
EJDE-2015/226
ENERGY DECAY
11
[11] M. Nakao; Stabilization of local energy in an exterior domain for the wave equation with a
localized dissipation, Journal of Differential Equations, 148 (1998), 388-406.
[12] M. Nakao; Energy decay for the linear and semilinear wave equations in exterior domains
with some localized dissipations, Mathematische Zeitschrift, 238 (2001), 781-797.
[13] P. F. Yao; Energy decay for the Cauchy problem of the linear wave equation of variable
coefficients with dissipation, Chin. Ann. of Math., ser. B, 31(1) (2010), 59-70.
[14] P. F. Yao; Observatility inequality for exact controllability of wave equations with variable
coefficients, SIAM J. Control Optim. 37 (1999), 1568-1599.
[15] P. F. Yao; Modeling and control in vibrational and structrual dynamics, A differential geometric approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series,
CRC Press, 2011.
[16] E. Zuazua; Exponential decay for the semilinear wave equation with locally distributed damping, Comm. PDE. 15 (1990), 205-235.
[17] E. Zuazua; Exponential decay for the semilinear wave equation with localized damping in
unbounded domains, J. Math. Pure et Appl. 70 (1992), 513-529.
Jieqiong Wu (corresponding author)
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
E-mail address: jieqiong@sxu.edu.cn
Fei Feng
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
E-mail address: fengfei599@126.com
Shugen Chai
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
E-mail address: sgchai@sxu.edu.cn